%I #7 Feb 06 2024 08:14:26

%S 0,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,

%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

%U 1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1

%N a(n) = sqrt(A369935(n)).

%H Amiram Eldar, <a href="/A369936/b369936.txt">Table of n, a(n) for n = 1..10000</a>

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (k * (d(k) - d(k-1)) / A357016 = 1.05307232234978965366..., where d(k) = Product_{p prime} (1 - 1/p^2 + Sum_{i=2..k} (1/p^(i^2)-1/p^(i^2+1))) for k >= 1, and d(0) = 0.

%t squareQ[n_] := IntegerQ[Sqrt[n]]; f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, squareQ], Sqrt[Max @@ e], Nothing]]; f[1] = 0; Array[f, 150]

%o (PARI) lista(kmax) = {my(e, q); print1(0, ", "); for(k = 2, kmax, e = factor(k)[, 2]; q = 1; for(i = 1, #e, if(!issquare(e[i]), q = 0; break)); if(q, print1(sqrtint(vecmax(e)), ", "))); }

%Y Cf. A357016, A369935.

%K nonn,easy

%O 1,12

%A _Amiram Eldar_, Feb 06 2024